QUESTIONSQUESTIONS

1. The definition of "1 atmosphere" is 1013 hPa, the average atmospheric pressure at sea level. But when we computed the mass of the atmosphere, we used a mean atmospheric pressure of 984 hPa. Why?

2. Torricelli used mercury for his barometer, but a column barometer could be constructed using any fluid, with the height of the column measuring atmospheric pressure given by h = P/fluidg

At sea level, with mercury fluid= 13.6 g cm-3 h = 76 cm

with water fluid= 1.0 g cm-3 h = 1013 cm

Now what about using air as a fluid?

with air fluid= 1.2 kg m-3 h = 8.6 km

which means that the atmosphere should extend only to 8.6 km, with

vacuum above! WHAT IS THE FLAW IN THIS REASONING?

CHAPTER 3: SIMPLE MODELSCHAPTER 3: SIMPLE MODELS

Define problem of interest

Design model; make assumptions needed to simplify equations and make them solvable

Evaluate model with observations

Apply model: make hypotheses, predictions

Improve model, characterize its error

The atmospheric evolution of a species X is given by the continuity equation

This equation cannot be solved exactly need to construct model (simplified representation of complex system)

Design observational system to test model

[ ]( [ ])X X X X

XE X P L D

t

U

local change in concentration

with time

transport(flux divergence;U is wind vector)

chemical production and loss(depends on concentrations of other species)

emissiondeposition

TYPES OF SOURCESTYPES OF SOURCESNatural Surface: terrestrial and marine

highly variable in space and time, influenced by season, T, pH, nutrients…eg. oceanic sources estimated by measuring local supersaturation in water and using a model for gas-exchange across interface =f(T, wind velocity….)

Natural In situ: eg. lightning (NOx) N2 NOx, volcanoes (SO2, aerosols) generally smaller than surface sources on global scale but important b/c material is injected into middle/upper troposphere where lifetimes are longer

Anthropogenic Surface:eg. mobile, industry, fires good inventories for combustion products (CO, NOx, SO2) for US and EU

Anthropogenic In situ:eg. aircraft, tall stacks

Secondary sources: tropospheric photochemistry

Injection from the stratosphere: transport of products of UV dissociation (NOx, O3) transported into troposphere (strongest at midlatitudes, important source of NOx in the UT)

TYPES OF SINKSTYPES OF SINKS

Wet Deposition: falling hydrometeors (rain, snow, sleet) carry trace species to the surface

• in-cloud nucleation (depending on solubility)• scavenging (depends on size, chemical composition)Soluble and reactive trace gases are more readily removedGenerally assume that depletion is proportional to the conc (1st order loss)

Dry Deposition: gravitational settling; turbulent transportparticles > 20 µm gravity (sedimentation)particles < 1 µm diffusion rates depend on reactivity of gas, turbulent transport, stomatal resistance and together define a deposition velocity (vd)

In situ removal:chain-terminating rxn: OH●+HO2● H2O + O2

change of phase: SO2 SO42- (gas dissolved salt)

d d xF v C Typical values vd:Particles:0.1-1 cm/sGases: vary with srf and chemical nature (eg. 1 cm/s for SO2)

RESISTANCE MODEL FOR DRY DEPOSITIONRESISTANCE MODEL FOR DRY DEPOSITION

Deposition Flux: Fd = -vdC Vd = deposition velocity (m/s)C = concentration

C3

C2

C1

C0=0

Aerodynamic resistance = ra

Quasi-laminar layer resistance = rb

Canopy resistance = rc

For gases at steady state can relate overall flux to the concentration differences and resistances across the layers:

Use a resistance analogy, where rT=vd-1

3 2 1 0 3 02 1

a b c T

3 b 1

a

3 b c

a

C C C C C CC CF

r r r r

C (Fr C )

r

C (Fr Fr )

r

3

a b c

1d a b c

CF

r r r

v r r r

00

For particles, assume that canopy resistance is zero (so now C1=0), and need to include particle settling (settling velocity=vs) which operates in parallel with existing resistances. End result:

d sa b a b s

1v v

r r r r v

Reference: Seinfeld & Pandis, Chap 19

ONE-BOX MODELONE-BOX MODEL

Inflow Fin Outflow Fout

XE

EmissionDeposition

D

Chemicalproduction

P L

Chemicalloss

Atmospheric “box”;spatial distribution of X within box is not resolved

out

Atmospheric lifetime: m

F L D

Fraction lost by export: out

out

Ff

F L D

Lifetimes add in parallel: export chem dep

1 1 1 1outF L D

m m m

Loss rate constants add in series:export chem dep

1k k k k

Mass balance equation: sources - sinks in out

dmF E P F L D

dt

(because fluxes add linearly)

(turnover time)

Flux units usually[mass/time/area]

ASIDE: LIFETIME VS RADIOACTIVE HALF-LIFEASIDE: LIFETIME VS RADIOACTIVE HALF-LIFE

Both express characteristic times of decay, what is the relationship?

1/2

1/2

[ ]( ) [ ](0)

1[ ]( ) [ ](0)

2ln(2)

0.7

ktA t A e

A t A

tk

½ life:

EXAMPLE: GLOBAL BOX MODEL FOR COEXAMPLE: GLOBAL BOX MODEL FOR CO22

reservoirs in PgC, flows in Pg C yrreservoirs in PgC, flows in Pg C yr-1-1

IPCC [2001]

atmospheric content (mid 80s)

= 730 Pg C of CO2

annual exchange land = 120 Pg C yr-1

annual exchange ocean= 90 Pg C yr-1

Human Perturbation

(now ~816 PgCO2)

SPECIAL CASE: SPECIAL CASE: SPECIES WITH CONSTANT SOURCE, 1SPECIES WITH CONSTANT SOURCE, 1st st ORDER SINKORDER SINK

Steady state solution (dm/dt = 0)

Initial condition m(0)

Characteristic time = 1/k for• reaching steady state• decay of initial condition

If S, k are constant over t >> , then dm/dt 0 and mS/k: "steady state"

( ) (0) (1 )kt ktdm SS km m t m e e

dt k

TWO-BOX MODELTWO-BOX MODELdefines spatial gradient between two domainsdefines spatial gradient between two domains

m1m2

F12

F21

Mass balance equations: 11 1 1 1 12 21

dmE P L D F F

dt

If mass exchange between boxes is first-order:

11 1 1 1 12 1 21 2

dmE P L D k m k m

dt

system of two coupled ODEs (or algebraic equations if system is assumed to be at steady state)

(similar equation for dm2/dt)

Illustrates long time scale for interhemispheric exchange; can use 2-box model to place constraints on sources/sinks in each hemisphere

TWO-BOX MODELTWO-BOX MODEL(with loss)(with loss)

m1 m2

T

Lifetimes:1 2 1 2

1 2 01 2 1 2

m m m m

T S S S S

If at steady state sinks=sources, so can also write:

S1 S2

Q

1 2 1 21 2 0

m m m m

Q T Q

Now if define: α=T/Q, then can say that: 1 2o

Maximum α is 1 (all material from reservoir 1 is transferred to reservoir 2), and therefore turnover time for combined reservoir is the sum of turnover times for individual reservoirs. For other values of α, the turnover time of the combined reservoir is reduced.

mo = m1+m2

EULERIAN EULERIAN RESEARCH MODELS SOLVE MASS BALANCE RESEARCH MODELS SOLVE MASS BALANCE

EQUATION IN 3-D ASSEMBLAGE OF GRIDBOXES EQUATION IN 3-D ASSEMBLAGE OF GRIDBOXES

Solve continuity equation for individual gridboxes

• Models can presently afford ~ 106 gridboxes

• In global models, this implies a horizontal resolution of 100-500 km in horizontal and ~ 1 km in vertical

• Drawbacks: “numerical diffusion”, computational expense

The mass balance equation is then the finite-difference approximation of the continuity equation.

EULERIAN MODEL EXAMPLEEULERIAN MODEL EXAMPLE

Summertime Surface Ozone Simulation

[Fiore et al., 2002]

Here the continuity equation is solved for each 2x2.5 grid box.They are inherently assumed to be well-mixed

IN EULERIAN APPROACH, DESCRIBING THE IN EULERIAN APPROACH, DESCRIBING THE EVOLUTION OF A POLLUTION PLUME REQUIRES EVOLUTION OF A POLLUTION PLUME REQUIRES

A LARGE NUMBER OF GRIDBOXES A LARGE NUMBER OF GRIDBOXES

Fire plumes oversouthern California,25 Oct. 2003

A Lagrangian “puff” model offers a much simpler alternative

PUFF MODEL: FOLLOW AIR PARCEL MOVING WITH WINDPUFF MODEL: FOLLOW AIR PARCEL MOVING WITH WIND

[X](xo, to)

[X](x, t)

wind

In the moving puff,

…no transport terms! (they’re implicit in the trajectory)

Application to the chemical evolution of an isolated pollution plume:

[X]

[X]b

In pollution plume,

[ ]d XE P L D

dt

[ ]([ ] [ ] )dilution b

d XE P L D k X X

dt

COLUMN MODEL FOR TRANSPORT ACROSS COLUMN MODEL FOR TRANSPORT ACROSS URBAN AIRSHEDURBAN AIRSHED

Temperature inversion(defines “mixing depth”)

Emission E

In column moving across city,[ ]

[ ]d X E k

Xdx Uh U

[X]

L0 x

[ ][ ]

[ ]

[ ]

d X Ek X

dt hd X dx

dx dtd X

Udx

( )/[ ]( ) [ ]( ) k x L UX x X L e

/[ ]( ) 1 kx UEX x e

hk [ ]( ) 0X x

Solution:

LAGRANGIAN LAGRANGIAN RESEARCH MODELS FOLLOW RESEARCH MODELS FOLLOW LARGE NUMBERS OF INDIVIDUAL “PUFFS”LARGE NUMBERS OF INDIVIDUAL “PUFFS”

C(x, to)Concentration field at time t

defined by n puffs

C(x, tot

Individual puff trajectoriesover time t

ADVANTAGES OVER EULERIAN MODELS:• Computational performance (focus puffs on region of interest)• No numerical diffusion

DISADVANTAGES:• Can’t handle mixing between puffs can’t handle nonlinear processes• Spatial coverage by puffs may be inadequate

FLEXPART: A LAGRANGIAN MODELFLEXPART: A LAGRANGIAN MODEL

[Cooper et al., 2005]

Retroplume (20 days): Trinidad Head, Bermuda

But no chemistry, deposition, convection here

x

Emissions Map (NOx)

=

Region of Influence